= First-order photon trajectory in scalar perturbations
{title2=$x^i=e^i(\eta-\eta_0)+\delta x^i$}
For reception at the origin with unit propagation direction $e$, the <null geodesic> condition fixes $v(\eta_0)=(1+2\Psi_0)e$. The perturbative spatial acceleration is $x^{i\prime\prime}=2e^i d\Psi/d\eta-2(\delta^{ij}-e^ie^j)\partial_j\Psi$. Its integrated trajectory is $x^i=e^i(\eta-\eta_0)+2e^i\int_{\eta_0}^{\eta}\Psi\,d\eta^{\prime}-2\int_{\eta_0}^{\eta}(\eta-\eta^{\prime})(\delta^{ij}-e^ie^j)\partial_j\Psi\,d\eta^{\prime}$. Potentials in these already first-order integrals can be evaluated on the straight background ray.
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